Extension dimension for paracompact spaces

Abstract

We prove existence of extension dimension for paracompact spaces. Here is the main result of the paper: \proclaim{Theorem} Suppose X is a paracompact space. There is a CW complex K such that {a.} K is an absolute extensor of X up to homotopy, {b.} If a CW complex L is an absolute extensor of X up to homotopy, then L is an absolute extensor of Y up to homotopy of any paracompact space Y such that K is an absolute extensor of Y up to homotopy. proclaim The proof is based on the following simple result (see 1.6). \proclaim{Theorem} Suppose X be a paracompact space and $f:A\to Y$ is a map from a closed subset A of X to a space Y. f extends over X if Y is the union of a family $\{Y_s\}_{s\in S}$ of its subspaces with the following properties: {a.} Each $Y_s$ is an absolute extensor of X, {b.} For any two elements s and t of S there is $u\in S$ such that $Y_s\cup Y_t\subset Y_u$, {c.} $A=\bigcup\limits_{s\in S} \int_A(f^{-1}(Y_s))$. proclaim That result implies a few well-known theorems of classical theory of retracts which makes it of interest in its own.